It may get wet if it is a soaking rain but should dry out. My main concern would be water getting into the house as tar paper does not seal completely.

Water getting into the house is my concern also. The contractor I have doing our roof is having his crew take off the entire roof before they shingle it. Half of it is now just tar paper and he said it will take 2 more days to get the rest off (they are working very slow!). In the meantime-we got a hard, long rain last night. They probably won't come today becuase more rain is expected. In fact they are calling for more rain the rest of the week off and on. I'm guessing they won't get to the shingles until first of the week at the earliest.
Should I be worried then? This is a house we're redoing and all the drywall, ceilings, etc. is new inside.

If laid out properly, the tar paper should suffice as long as it is intact. With a strong enough wind gust, it could tear and leave a gap in the coverage. Protecting the tar paper with a tarp might be good insurance.

I was remiss in not mentioning that as well, a large tarp would suffice perfectly.

We haven't had any bad winds with the rain and the tar paper seems to be intact, but in the morning when it is wet, it is all rippled. Is this OK? It seems to flatten back out once it dries out.

Finished house or new construction?

Tarps are normally used on a finished house for protection, while in a new house there should not be anything that would be damaged.

It probably would have been better if you waited
until the roof was done before you put up new
sheetrock, but maybe you didn't know the roof
was bad at that time. I believe a good roofing
contractor would have completed half the roof
before doing the tear-off on the other side.
Ask him about that, I'm curious. Also, does
he have insurance, in case your sheetrock got
ruined?

That seems to be awfully slow work. It's a little alarming, actually, as to how slow the project is going. He's getting paid by the job, not the hour, right?

There's nothing wrong with tar paper as a roof surface for a little while. The danger comes with the guys walking on it or with wind blowing rain under the laps. For asphalt roofing, it does need to be as flat as possible when installing the shingles over it. If it's getting quite a bit of curl/wrinkle, I'm guessing he used 15 lb. paper rather than 30 lb.

Me personally, after he's gotten to the end and the roof is all off, I'd want the 15 lb. taken up and 30 lb. installed before shingling began. That might be an hour's worth of time and another $150 in materials for a proper roofing crew. And, it's overkill according to most manufacturers. I'm all about overkill when it's that darn cheap to do though.